BRAIDED COMMUTATIVE ALGEBRAS OVER QUANTIZED ENVELOPING ALGEBRAS

We produce braided commutative algebras in braided monoidal categories by generalizing Davydov’s full center construction of commutative algebras in centers of monoidal categories. Namely, we build braided commutative algebras in relative monoidal centers ZℬC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal{Z}}_{\mathrm{\mathcal{B}}}\left(\mathcal{C}\right) $$\end{document} from algebras in ℬ-central monoidal categories C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{C} $$\end{document}, where ℬ is an arbitrary braided monoidal category; Davydov’s (and previous works of others) take place in the special case when ℬ is the category of vector spaces VectK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbf{Vect}}_{\mathbbm{K}} $$\end{document} over a field K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbbm{K} $$\end{document}. Since key examples of relative monoidal centers are suitable representation categories of quantized enveloping algebras, we supply braided commutative module algebras over such quantum groups. One application of our work is that we produce Morita invariants for algebras in ℬ-central monoidal categories. Moreover, for a large class of ℬ-central monoidal categories, our braided commutative algebras arise as a braided version of centralizer algebras. This generalizes the fact that centers of algebras in VectK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbf{Vect}}_{\mathbbm{K}} $$\end{document} serve as Morita invariants. Many examples are provided throughout.


Introduction
Let k be a field and note that all algebraic structures in this manuscript are k-linear. The purpose of this work is to systematically produce and study braided commutative algebras (or, commutative algebras, for short) in a certain well-behaved class of braided monoidal categories. This is achieved by generalizing Davydov's full center construction in [Dav10,Dav12] for commutative algebras in centers of monoidal categories ZpCq, which was built on works of Fröhlich-Fuchs-Runkel-Schweigert [FFRS06] and of Kong-Runkel [KR08] in their studies of algebras in modular tensor categories.
Braided commutative algebras are interesting mathematically for several reasons. For instance, they can be used to provide natural examples of bialgebroids, which are generalizations of kbialgebras with a base algebra possibly larger than k. Namely, for a bialgebra H and an algebra A in the (braided monoidal) category of H-Yetter-Drinfeld modules, we have that A is commutative if and only if the smash product algebra A H admits the structure of a bialgebroid with base algebra A [BM02, Theorem 4.1], [Lu96,Theorem 5.1].
Commutative algebras in braided monoidal categories also have applications to physics. For instance, extended chiral algebras in rational conformal field theory (RCFT) arise as commutative algebras in modular tensor categories [FRS02, Section 5.5]. These algebras were shown to be Morita invariants in modular tensor categories, and are used to prove that in two-dimensional RCFT there cannot be several incompatible sets of boundary conditions for a given bulk theory [KR08].
Moreover, commutative algebras in braided monoidal categories C have been used to classify certain extensions of vertex operator algebras; see [HKL15] and [HK04] for more details.
We anticipate that our construction of braided commutative algebras here will have similar and new implications both in mathematics and physics. For now, note that we deliver a supply of commutative algebras in (braided monoidal) representation categories of quantized enveloping algebras, a result that extends beyond work in [Dav10,Dav12] as we discuss below.
In this work, we build commutative algebras in relative monoidal centers Z B pCq [Definition 3.3], which is a class of braided monoidal categories studied by the first author [Lau15,Lau20] (motivated by [BD98,Maj99], and related to [Müg03,Definition 2.6]; see also [DNO13,Section 4]). Here, B is a braided monoidal category, and C is a monoidal category that is B-central [Definition 3.1]. For instance, when B is the category of k-vector spaces Vect k , we have that Z B pCq is the usual monoidal center ZpCq of C [JS91,Maj91]. In general, Z B pCq is a proper subcategory of ZpCq [Proposition 3.5, Example 3.11]. Analogous to Davydov's full center construction for commutative algebras in ZpCq [Dav10], we show that if there exists a functor that is a right adjoint to the forgetful functor Z B pCq Ñ C, then it is lax monoidal [Lemma 3.12] (so it sends algebras in C to algebras in Z B pCq). Our method for producing commutative algebras in Z B pCq is called the B-center construction; see Section 3.3 and Theorem 3. 24.
Key examples of Davydov's work occur when C " H-Mod, the category of modules 1 over a Hopf algebra H; if H is finite-dimensional, then ZpCq is equivalent to the category of modules over the Drinfeld double of H [Dri86,Dri87]. Now we construct a larger class of commutative algebras in braided monoidal categories, including commutative algebras in representation categories of braided Drinfeld doubles [Lau15] (or, of double bosonizations [Maj99]), including those in our title. In particular, take g a semisimple Lie algebra over k with positive/negative nilpotent part n`{´and positive/negative Borel part g`{´. Then, ‚ Z B pCq » U q pgq-Mod lfw , the category of locally n`-finite weight modules over a quantized enveloping algebra of g over kpqq, for q a generic variable, when ‚ C " U q pg´q-Mod w , the category of weight modules over the negative Borel part, and ‚ B " K-CoMod, where K " U q phq is the quantized enveloping algebra of the Cartan subalgebra h of g, also realized as a group algebra of a lattice; (:) see [Lau20, Section 4.3]. One can also work with small quantum groups, cf. [Lau20, Section 4.4]: ‚ Z B pCq » u q pgq-Mod, the category of modules over a finite-dimensional quantized enveloping algebra of g, for q a root of unity, when ‚ C " u q pg´q-Mod, and ‚ B " K-Mod, where K is a group algebra of a certain finite abelian group.
(;) Let us consider a setting more general than p:, ;q as follows. Take: We also recall in Definition 5.2 and Proposition 5.3 that there is a functor Φ from H H YDpBq to the category of representations of the braided Drinfeld double Drin K pH˚, Hq . Here, Drin K pH˚, Hq is the usual Drinfeld double of H when K " k and H is a finite-dimensional Hopf algebra. In a special case, U q pgq -Drin Uqphq pU q pn`q, U q pn´qq as shown in [Maj99,Section 4], see also [Lau19, Section 3.6].
We verify the functor R B from (1.1) exists in setting (‹) [Theorem 3.13]. We also establish under the setting p‹q that the image of an algebra A in C under R B is a braided version of the centralizer algebra Cent l A H pAq Ψ´1 of A in A H [Theorem 4.5]. This is analogous to the main result of [Dav12] in the case B " Vect k .
Our main constructions and results are summarized in Figure 1 below for setting p‹q when C " H-ModpBq, although much of the work below holds for arbitrary B-central monoidal categories.  The B-center was discussed above after (1.1). The left center, considered initially in [VOZ98,Ost03,FFRS06] for a given braided monoidal category D, was used in [Dav10] to produce commutative algebras in D from algebras in D.
In comparison with [Dav10,Dav12], to achieve our constructions above we must use more involved techniques of graphical calculus sensitive to the order of crossing strands, since we work in braided monoidal categories B more general than Vect k . In any case, with B-centers we are able to produce Morita invariants of algebras in B-central monoidal categories as discussed below.
Theorem 3.26. Take C a B-central monoidal category, and algebras A, A 1 in C. Suppose that the categories of right modules over A and over A 1 in C are equivalent as left C-module categories. Then, the B-centers Z B pAq and Z B pA 1 q are isomorphic as commutative algebras in Z B pCq.
In reference to setting p;q, for instance, one can employ the theorem above to produce Morita invariants for u q pn´q-module algebras by using braided commutative u q pgq-module algebras.
In addition to Davydov's work [Dav10,Dav12] and the first author's work on comodule algebras over braided Drinfeld doubles [Lau19], our results have connections to several other articles in the literature. See several works on braided commutative algebras in Yetter-Drinfeld categories, including [CFM99,CVOZ94,CW94]. See also work of Montgomery-Schneider [MS01], of Cline [Cli19], and of Kinser and the second author [KW16] on extending module algebras over Taft algebras to those over their Drinfeld double and over u q psl 2 q. On another related note, Etingof-Gelaki realized the representation category of a small quantum group u q pgq as the monoidal center of a representation category of a certain quasi-Hopf algebra A q pgq [EG09]. This paper is organized as follows. We discuss categorical preliminaries in Section 2, including the left center construction; we also introduce braided centralizer algebras there. Next, we introduce and study the B-center construction and the functor R B in Section 3, and we also verify Theorems 3.24 and 3.26 and discuss the special case when C " B, which is a monoidal category central over itself. Then, we restrict our attention to setting p‹q in Section 4 and show that algebra images under R B are braided centralizer algebras. Towards constructing examples for the material in Sections 2-4, we discuss braided Drinfeld doubles and Heisenberg doubles in Section 5. Finally, we provide examples of our results for braided commutative algebras in the representation categories of the small quantum group u q psl 2 q in Section 6, and of the Sweedler Hopf algebra T 2 p´1q in Section 7. We discuss how to generalize the detailed work in Section 6 to U q pgq and u q pgq in Section 8, and we end by listing several directions for further investigation there.

Categorical preliminaries
In Section 2.1, we first set up notation and conventions that we will use throughout this work. Next, we recall terminal objects and the comma category in Section 2.2, and then discuss in Section 2.3 the left and right center construction that produces commutative algebras in braided monoidal categories from algebras in such categories. Finally, we introduce and study braided versions of centralizer algebras in Section 2.4.
2.1. Notation and conventions. All categories in this work are abelian, complete under arbitrary countable biproducts, and enriched over the category of k-vector spaces Vect k . The reader may wish to refer to [EGNO15] or [TV17] for further background information.
Throughout, C " pC, bq denotes a monoidal category; later in Section 3.1, C will be B-central in the sense of Definition 3.1 for a braided monoidal category B. The tensor unit of each monoidal category is denoted by I. We usually omit the associativity and unitarity isomorphisms for monoidal categories which is justified by MacLane's coherence theorem.
Unless stated otherwise, we assume that all monoidal functors F : C Ñ C 1 are strong monoidal, i.e., there exists a natural isomorphism that is compatible with the associativity constraints of C and C 1 , and so that F pI C q -I C 1 .
We denote by AlgpCq the category of algebras pA, m, uq in C; here, A is an object of C with associative multiplication m : A b A Ñ A and unit u : I Ñ A. As usual, given an algebra A in C, a left A-module is a pair pV, a V q for V an object in C and a V : A morphism of A-modules pV, a V q Ñ pW, a W q is a morphism V Ñ W in C that intertwines with a V and a W . This way, we define the category A-ModpCq of left A-modules in C. Analogously, we define Mod-ApCq, the category of right A-modules in C.
Moreover, we reserve D to be an arbitrary braided monoidal category with braiding Ψ D , or Ψ if D is understood. In this case, we can consider bialgebras and Hopf algebras in D, and we assume that all such Hopf algebras in our work have an invertible antipode.
If pA, mq is an algebra in pD, Ψq, then A is braided commutative (or, commutative) if mΨ " m (or, equivalently, if mΨ´1 " m) as morphisms in D. We denote the full subcategory of AlgpDq of commutative algebras by ComAlgpDq. The braided opposites of pA, mq are A Ψ :" pA, mΨq and A Ψ´1 :" pA, mΨ´1q with respect to the braiding and inverse braiding of D, respectively. If A is braided commutative, then A " A Ψ " A Ψ´1 .
We use the graphical calculus as in [Lau19], similar to that used in [Maj94], for computations in (braided) monoidal categories D. The braiding in D is denoted by Moreover, for a Hopf algebra H :" pH, m, u, ∆, ε, Sq in D, we denote: Combining these symbols, we can display all axioms of a Hopf algebra in D (see, e.g., [Lau19, Equations 1.2-1.9]). For example, the bialgebra condition becomes A left H-action on V P D is denoted by Moreover, the H-module structure a V bW on the tensor product V b W of left H-modules V , W becomes: Now for H a Hopf algebra in a braided monoidal category pD, Ψq, we have via (2.1) that H-ModpDq is a monoidal category. Similarly, AlgpDq is a monoidal category: The tensor product of two algebras A, B in D is given by pA b B, m AbB , u AbB q, where 2.2. Terminal objects and the comma category. We record some useful results about terminal objects and the comma category that we will use below. Recall that an object T of C is terminal if, for any object X P C, there exists a unique morphism X Ñ T in C. If a terminal object exists, then it is unique up to unique isomorphism. By considering the morphisms T b T Ñ T (multiplication) and I Ñ T (unit) in C, one can verify the associativity and unit axioms to obtain the following fact.
Lemma 2.2. The terminal object of a (braided) monoidal category is a (commutative) algebra.
Moreover, we get the following fact from the uniqueness of morphisms to the terminal object.
Lemma 2.3. If A is an algebra in a monoidal category C, then the unique morphism A Ñ T to the terminal object is a morphism of algebras in C.
We will also need to use the next construction later. Take two monoidal categories C and C 1 , a monoidal functor F : C Ñ C 1 , and an object A P C 1 . Then, the comma category FÓA is the category of pairs pX, xq with X P C and x : F pXq Ñ A in C 1 , with morphisms being morphisms of the first components that are compatible with the morphisms of the second components. If F is monoidal and A P AlgpC 1 q, then FÓA is also monoidal with Definition 2.4. Let pA, mq be an algebra in pD, Ψq. The left center C l pAq Ñ A of A is the terminal object in the category of morphisms γ : Z Ñ A such that the following diagram commutes.
Equivalently, it is defined as the maximal subobject C l pAq of A such that mΨ C l pAq,A " m as maps from C l pAq b A to A in D.
Similarly, we define the right center C r pAq Ñ A using Ψ´1 instead of Ψ.
Proposition 2.6. [Dav10, Proposition 5.1] The left center C l pAq has an algebra structure in D, unique up to unique isomorphism of algebras, such that C l pAq Ñ A is a morphism in AlgpDq. In addition, C l pAq P ComAlgpDq. Similarly, C r pAq P ComAlgpDq.

Left and right centralizers.
We generalize the left and right center constructions in the previous section as follows.
Definition 2.7. Let pA, mq be an algebra, S be an object, and φ : S Ñ A be a morphism in pD, Ψq. The left centralizer Cent l A pSq Ñ A of A is a terminal object in the category of morphisms γ : C Ñ A such that the following diagram commutes.
Similarly, we define the right centralizer Cent r A pSq Ñ A using Ψ´1 instead of Ψ.
Example 2.9. Let A be an algebra in D.
(1) Let S " A and φ " Id A . In this case, Cent l A pAq " C l pAq, and Cent r A pAq " C r pAq. (2) Let S " I, and let φ " u A : I Ñ A be the unit of A. In this case, Cent l A pIq " Cent r A pIq " A.
Definition 2.10. Let A be an algebra, S be an object, and φ : S Ñ A be a morphism in D. We denote by C l A pSq the category consisting of ‚ objects which are pairs pC, γq, where C is an object and γ : C Ñ A is a morphism in D that make Diagram 2.8 commute; with ‚ morphisms pC, γq Ñ pC 1 , γ 1 q that are morphisms f : C Ñ C 1 in D such that the diagram below commutes.
(2.11) From Example 2.9(1), we denote C l A pAq by C l pAq. Now the left centralizer Cent l A pSq is the terminal object of C l A pSq, and the left center C l pAq is the terminal object of C l pAq. Similar to Proposition 2.6, we have the result below.
Proposition 2.12. The left centralizer Cent l A pSq has the structure of an algebra in D, which is unique up to (unique) algebra isomorphism, such that Cent l A pSq Ñ A is morphism in AlgpDq.
Proof. By the discussion above and the material in Section 2.2, it suffices to show that C l A pSq is a monoidal category. Towards this, take pC 1 , γ 1 q and pC 2 , γ 2 q in C l A pSq and define pC 1 , γ 1 q b pC 2 , γ 2 q :" pC 1 b C 2 , m A pγ 1 b γ 2 qq.
, , Unlike Proposition 2.6 (for S " A), neither Cent l A pSq or Cent r A pSq necessarily belongs to ComAlgpDq: To see this use Example 2.9(2). In any case, consider the example below.
Example 2.13. Suppose that D is the category of modules over a quasi-triangular Hopf algebra (so it comes equipped with a fiber functor to Vect k and objects have elements). Then, for any subobject S Ă A, the left centralizer Cent l A pSq is isomorphic to the following subalgebra of A: Cent l A pSq " ta P A | mpa b sq " mΨpa b sq, @s P Su .

The functor R B and the B-center
We present the main results of our work in this section. First, we discuss background material on B-central monoidal categories and relative monoidal centers Z B pCq adapted from [Lau20] in Section 3.1. In Section 3.2, we study the right adjoint R B to the forgetful functor Z B pCq Ñ C, and show that it exists when Z B pCq is a Yetter-Drinfeld category H H YDpBq in B [Theorem 3.13]; the lax monoidal property of R B is also discussed. Next, in Section 3.3, we generalize Davydov's full center construction [Dav10, Section 4] to the B-central setting, thus producing braided commutative algebras Z B pAq in Z B pCq from algebras A in C [Proposition 3.22]. In Section 3.4, we show that for A P AlgpCq we have an isomorphism Z B pAq -C l pR B pAqq of algebras in ComAlgpZ B pCqq if R B exists and its adjunction counit is an epimorphism [Theorem 3.24]; this generalizes [Dav10, Theorem 5.4]. In Section 3.5, we establish that B-centers serve as Morita invariants for algebras in B-central monoidal categories [Theorem 3.26]. Finally, in Section 3.6, we restrict our attention to the case when C is braided and central over itself and present the results of the previous subsections in this setting.
3.1. B-central monoidal categories and relative monoidal centers. Recall B :" pB, Ψq denotes a braided monoidal category. From now on C will be a monoidal category of the kind below.
Definition 3.1. We say that a monoidal category C is B-central monoidal if it comes equipped with a faithful monoidal functor, called the central functor,  (1) By the assumptions in Section 2.1, all monoidal categories in this work are Vect k -central.
(2) We have that B is B-central with T " Id B and σ " Ψ.
(3) Take H a Hopf algebra in B, and consider the category C :" H-ModpBq of left H-modules in B. This category is monoidal: Take pV, a V : H b V Ñ V q and pW, a W : H b W Ñ W q in C and we get that pV b W, a V bW q P C with a V bW defined as in Equation 2.1. Moreover, C is B-central monoidal with T giving an object of B the structure of a trivial H-module in B (via the counit of H), and σ :" TpΨ FpV q,B q for all V P C and B P B.
Example 3.2(3) will play a crucial role in our work later. Next, we define the relative monoidal center of a B-central monoidal category C.
Definition 3.3. [Lau20, Definition 3.32, Propositions 3.33 and 3.34] The relative monoidal center Z B pCq of a B-central monoidal category C is a braided monoidal category consisting of pairs pV, cq, where V is an object of C, and c :" is a natural isomorphism of half-braidings satisfying the two conditions below: (i) [tensor product compatibility] for any X, Y P C the following diagram commutes: Here, the monoidal structure is given by and the braiding is given by Ψ pV,c V,´q ,pW,c W,´q " c V,W .
Remark 3.4. The relative center Z B pCq is the Müger centralizer [Müg03, Definition 2.6] of the set of objects pTpBq, σ´1 B,V q in ZpCq.
In [Lau20], the relative monoidal center Z B pCq is defined (equivalently) as the monoidal category of B-balanced endofunctors G of the regular C-bimodule category C; here, using composition as the tensor product, i.e., G b G 1 " G 1 G, this is also a braided monoidal category.
Relative monoidal categories have the following properties, some of which hold by definition.
Proposition 3.5. Let C be a B-central monoidal category over a braided monoidal category B.
(1) If C " B, then Z B pBq is isomorphic to B as braided monoidal categories.
Note that the forgetful functor Z B pCq Ñ C does not necessarily have a right adjoint, but we show later in Lemma 3.12 that if a right adjoint exists, then it is lax monoidal. In particular, such a right adjoint exists for the B-central monoidal category H-ModpBq from Example 3.2(3); see Theorem 3.13. Toward this result, consider the following explicit description of the relative monoidal center of H-ModpBq in terms of crossed or Yetter-Drinfeld modules.
Definition 3.6. [Bes97, BD98] Take a Hopf algebra H in B, and take C :" H-ModpBq from Example 3.2(3) with left H-action in B denoted by a :" a V :" a H V . Then the category H H YDpBq of H-Yetter-Drinfeld modules in B consists of objects pV, a, δq where pV, aq P C with left H-coaction in B denoted by δ :" δ V :" δ H V , subject to compatibility condition: Given two objects pV, a V , δ V q and pW, a W , δ W q in H H YDpBq, their tensor product is given by Here, the braiding of H H YDpBq is given by Next, we illustrate Proposition 3.5(3) for C :" H-ModpBq on the level of objects below.
Example 3.11. Let K be a quasi-triangular Hopf algebra in Vect k with R-matrix R p1q b R p2q , and let H be a Hopf algebra in B " K-Mod. Then the smash product algebra H K is a Hopf algebra in Vect k (called bosonization or Radford biproduct) such that Hence, the essential image of this functor consists precisely of objects of H K H K YD isomorphic to those where the coaction δ restricted to K has the form δpvq " R p2q b a H K pR p1q b vq, i.e. is induced from the action a H K by using the universal R-matrix. This illustrates how a relative monoidal center Z B pCq is a proper subcategory of the monoidal center ZpCq when B is inequivalent to Vect k .

3.2.
A right adjoint to the forgetful functor. Let C be a B-central monoidal category. Consider the forgetful functor F B : Z B pCq Ñ C. In this section, we consider a general situation in which the forgetful functor F B has a right adjoint R B . In this case, we label the corresponding adjunction natural isomorphisms as follows: Lemma 3.12. Assume that F B : Z B pCq Ñ C has a right adjoint R B . Then the adjunction natural transformations α and β are monoidal, and R B is lax monoidal.
Proof. This follows from a general fact in category theory. The functor F B is strong monoidal, so it is oplax monoidal, and hence its right adjoint R B is lax monoidal [Kel74] (see also [nLa18]). A direct proof of these results is also given in [Dav10, Section 5]. Now assume that H is Hopf algebra in B, that C " H-ModpBq, and recall from Proposition 3.10 that Z B pCq » H H YDpBq. We construct a right adjoint R B to F B in the next result; this generalizes the construction from [CMZ97, Corollary 2.8] when B " Vect k (which is used crucially in [Dav12]). Moreover, the functor R B sends an object pV, a V q to the object pH b V, a R , δ R q with H-action a R and H-coaction δ R given by The lax monoidal structure is given by the morphism u : Proof. Checking that R B pV q is an object in the category of H-Yetter-Drinfeld modules in B is carried out using graphical calculus, especially since Sweedler notation cannot be employed easily for objects of B; this argument is quite similar to [Dav12, Proposition 5.1] for the case when B is symmetric monoidal. Here, the H-action and H-coaction on R B pV q is displayed as follows: We leave verification of the action condition a R pId H ba R q " a R pm b Id HbV q to the reader. The Yetter-Drinfeld condition, Equation 3.7, is verified by the following graphical calculation."

"
" " Here, the first and last equalities follow from (3.14). The second and third equality use coassociativity, the bialgebra axiom, and naturality of the braiding. The fourth equality follows from a computation using the antipode axioms and bialgebra condition while the fifth equality again uses naturality of the braiding and coassociativity. Functoriality of R B is clear by definition. To see that R B is right adjoint to F B , we present the unit α and counit β of the adjunction. For objects V of H-ModpBq and W of H H YDpBq, define A direct check verifies the adjunction axioms for α, β, and further one can check directly that α V is a morphism in H H YDpBq and β V a morphism in H-ModpBq. The lax monoidal structure is computed as in [Dav10, Section 5] as Using (3.15) and omitting associativity, we have that τ V,W : We have to verify associativity and unitarity squares for this lax monoidal structure, and these follow directly from the corresponding properties of H. It is also directly verified that u and τ V,W are indeed morphisms in H H YDpBq.
For any lax monoidal functor G : C Ñ D with A an algebra object in C, we get that GpAq is an algebra in D with product m GpAq " Gpm A qG A,A . Using this, we observe the following: Corollary 3.16. The lax monoidal functor R B induces the following functor on categories of algebra objects. Given an algebra A with product m A in H-ModpBq, the algebra R B pAq has product given by that is, given by the tensor product algebra structure on H b A in B. 3.3. The B-center. This section contains the categorical definition of the B-center, which is a direct generalization of the full center of Davydov's works [Dav10,Dav12] relative to a braided monoidal category B. Davydov's case corresponds to specializing B " Vect k .
Definition 3.19. Let A be an algebra in a B-central monoidal category C. The B-center of A is a pair (Z B pAq, ζ A ), where Z B pAq is an object in Z B pCq with half-braiding c Z B pAq,A and ζ A :" ζ B A : Z B pAq Ñ A is a morphism in C, which is terminal among pairs ppZ, c Z,A q, ζ : Z Ñ Aq in the comma category F B ÓA so that the following diagram commutes.
When B " Vect k , the pair pZpAq :" Z Vect k pAq, ζ Vect k A q is the full center of A (as in [Dav10,Dav12]).
The B-center of A is realized as a terminal object of the following braided monoidal category.
Definition 3.21. Let A be an algebra in C. We denote by Z B pAq the category consisting of ‚ pairs pZ, ζq with Z " pZ, cq an object in Z B pCq, and ζ : Z Ñ A a morphism in C, that make Diagram 3.20 commute; and ‚ morphisms pZ, ζq Ñ pZ 1 , ζ 1 q in Z B pCq that are morphisms f : Z Ñ Z 1 such that the diagram below commutes.
Given objects pZ, ζq, pZ 1 , ζ 1 q P Z B pAq, their tensor product is pZ b Z 1 , mpζ b ζ 1 qq, using the tensor product Z b Z 1 in Z B pCq, cf. [Dav10,Remark 4.2]. This makes Z B pAq a monoidal category.
The category Z B pAq is braided via which is a morphism in Z B pAq by commutativity of the outer diagram in The upper middle diagram commutes by naturality of c. We have the following result.
Proposition 3.22. The B-center Z B pAq of A P AlgpCq is a braided commutative algebra in Z B pCq and ζ A : Z B pAq Ñ A is a morphism of algebras in C.
Proof. It follows from Lemma 2.2 that pZ B pAq, ζ A q, being the terminal object of the braided monoidal category Z B pAq, is a commutative algebra in Z B pAq. Since the forgetful functor Z B pAq Ñ Z B pCq is a braided monoidal functor, we get that Z B pAq is a commutative algebra in Z B pCq. Moreover, the product m Z B pAq is a morphism in Z B pAq. So, and this condition means that ζ A is a morphism of algebras in C.
Corollary 3.23. For any algebra A in C, there is a unique morphism of algebras ξ A : Z B pAq Ñ ZpAq in ZpCq, which commutes with the respective morphisms of algebras to A.
Proof. Recall from Proposition 3.5(3) that Z B pCq is a braided monoidal subcategory of ZpCq. This implies that Z B pAq is a braided monoidal subcategory of ZpAq. Hence, Z B pAq is an algebra in ZpAq " Z Vect k pAq. By Lemma 2.3, we see that the unique morphism ξ A : Z B pAq Ñ ZpAq is one of algebras in ZpAq. In particular, ξ A is a morphism of algebras in ZpCq, which commutes with the respective morphisms to A in the sense that the following diagram commutes. Theorem 3.24. For a B-central monoidal category C, assume that there exists a right adjoint R B to the forgetful functor Z B pCq Ñ C, and that the counit is given by epimorphisms. Let A P AlgpCq.
Then, there is a canonical isomorphism of (commutative) algebras C l pR B pAqq -Z B pAq in Z B pCq.
Proof. Given the B-central set-up provided in previous sections, the proof of the theorem for the relative monoidal center Z B pCq is now analogous to Davydov's formal proof for ZpCq in [Dav10, Theorem 5.4]. The proof crucially uses the hypothesis that β A is an epimorphism.
3.5. Morita invariants. Next, we turn our attention to module categories over the monoidal categories discussed above. A left module category over a monoidal category C is a category M equipped with an bifunctor˚: CˆM Ñ M and natural isomorphisms for associativity The collection of C-module endofunctors of a C-module category M is a monoidal category and is denoted by E nd C pMq.
For an algebra A P C, recall that Mod-ApCq is the category of right modules over A in C. It is a left C-module category via X˚pM, ρq :" pX b M, Id X b ρq for all X P C and M P Mod-ApCq with structure morphism ρ : M b A Ñ M in C.
Definition 3.25. We say that two algebras A and A 1 in a monoidal category C are Morita equivalent if Mod-ApCq and Mod-A 1 pCq are equivalent as left C-module categories.
The above generalizes the notion of Morita equivalence for rings or for algebras in Vect k . We will establish the following result later in this section.
Theorem 3.26. Take C a B-central monoidal category, and let A and A 1 be algebras in C. Suppose that Mod-ApCq and Mod-A 1 pCq are equivalent as left C-module categories. Then, the B-centers Z B pAq and Z B pA 1 q are isomorphic as (commutative) algebras in Z B pCq. In particular, the B-center of an algebra in C is a Morita invariant.
This result is a generalization of [Dav10, Theorem 6.2 and Corollary 6.3] in the case when B " Vect k , and see the discussion in Remark 4.20 in the next section for an example of how it can be used in practice. For the proof of the theorem above, we need the next construction.
Definition 3.27. Take C a B-central monoidal category, and let M be a left C-module category. Consider the monoidal functor where L V : M Ñ M is the functor given by M Þ Ñ V˚M , and s is the collection of natural isomorphisms, for each X P C and M P M, given by Then the B-center of M is defined to be the terminal object in the comma category EÓI E nd C pMq .
Namely, Z B pMq is the terminal object amongst pairs ppZ, c Z,´q , zq for pZ, c Z,´q P Z B pCq and tz " z M : Z˚M Ñ M u MPM a natural transformation, such that for all X P C, M P M, the following diagram commutes.
In turn, it suffices to show that the comma category EÓI E nd C pMod-ApCqq used in Definition 3.27 is monoidally equivalent to the category Z B pAq from Definition 3.21. At this point, one can proceed exactly as in the proof of [Dav10, Theorem 6.2] using only the half-braidings of the full braided monoidal subcategory Z B pCq of ZpCq in order to finish the proof.
The converse of Theorem 3.26 holds when B " Vect k , with C a (braided monoidal) modular tensor category, and the algebras in C in question being simple and non-degenerate, by [KR08, Section 4.4]. So we ask: Question 3.28. In general, what conditions do we need on C, on B, and on algebras in C for a converse of Theorem 3.26 to hold?
We discuss the special setting of when C is braided next.
3.6. The case when C is braided. As mentioned in Example 3.2(2) and Proposition 3.5(1), we have that B is B-central, and that Z B pBq is isomorphic to B as braided monoidal categories. For instance, take C " k-ModpBq -this can be identified canonically with B and is isomorphic to Z B pBq as braided monoidal categories. Moreover in this case, there is a natural isomorphism R B " ùñ Id B , and the B-center of A P AlgpBq is given by Z B pAq -C l pR B pAqq -C l pAq as commutative algebras in Z B pBq by Theorem 3.24.
So, when C is braided, one does not need to work outside of C to get Morita invariants for algebras in C via Theorem 3.26, as C is isomorphic to its relative monoidal center. This is computationally more feasible than working with the full center construction of [Dav10]; see, for example, Remark 6.2. In particular, constructing Morita invariants of algebras in modular tensor categories was one of the motivations behind Davydov's work [Dav10] and other previous works [FFRS06, KR08] -now our construction of B-centers makes this goal more tractable computationally.

Connection to centralizer algebras
In this section, we restrict our attention to the situation where H is a Hopf algebra in a braided monoidal category B, and C " H-ModpBq. Take A P AlgpCq. We saw in Theorem 3.24 that the B-center Z B pAq can be computed as the left center of R B pAq, and we will now realize Z B pAq as a braided version of a centralizer algebra -see Theorem 4.5 below. We begin by discussing braided smash product algebras in Section 4.1. Then Theorem 4.5 is established in Section 4.2, and consequences of this result are provided in Section 4.3.
4.1. Braided smash product algebras. Consider the following terminology.
Moreover, the result below describes the category of modules over braided smash product algebras.
Next, we provide a preliminary result on braided smash product algebras. Proof. The action a and the coaction δ are defined from the action a R and coaction δ R in Theorem 3.13 by requiring that ϕ becomes a morphism of Yetter-Drinfeld modules. That is, Since a R , δ R are Yetter-Drinfeld compatible, a , δ are also Yetter-Drinfeld compatible. Theorem 4.5. For any algebra A in H-ModpBq, its B-center Z B pAq is isomorphic to Cent l A H pAq Ψ´1 as an algebra in B.
Before proving the theorem, we need the following lemma.
Lemma 4.6. Let A be an algebra in C " H-ModpBq. The left center C l pR B pAqq is the terminal object in the category of morphisms γ : or, equivalently, (4.8) We will only need (4.8) for the proof of Theorem 4.5, and using graphical calculus (4.8) iś This follows from the following series of equalities of morphisms from A b C to A H, which we display using graphical calculus:" Here, the first equality uses that ∆ preserves the unit, which acts by the identity. The second equality uses (4.9), while the third equality uses naturality of the braiding. Finally, the last equality uses the unit axioms. Pre-composing with Ψ A H,A H , we see that Equation 4.12 is equivalent to (4.14) This is Diagram 2.8 for the left centralizer of A inside of A H. Hence, ΓpC, γq is an object in C l A H pAq. Conversely, if pC 1 , ζq is an object in C l A H pAq, then pC 1 , ϕ´1ζq defines an object in C l pR B pAqq such that ΓpC 1 , ϕ´1ζq " pC 1 , ζq. This holds because after applying ϕ´1 to the computation in (4.13), we recover (4.9). Hence Γ is essentially surjective. As Γ is the identity on morphisms, it is fully faithful and therefore an equivalence of categories by Lemma 4.6.
Next, we show that Γ is a Ψ´1-opposite monoidal functor. Indeed, given two objects pC, γq and pC 1 , γ 1 q we have the following string of equalities of morphisms from C b C 1 to A H: γ´.
The first equality holds by (4.9). The second and fourth equality use naturality of the braiding, while the third equality uses the antipode axioms. This calculation gives that is an isomorphism in C l A H pAq. Finally, the equivalence Γ sends terminal objects to terminal objects. Using the discussion from Section 2.2, this implies that there is an isomorphism of algebras (4.15)φ : C l pR B pAqq Ñ Cent l A H pAq Ψ´1 such that the diagram Proof. Denote C 1 :" C l pR B pAqq, with multiplication m 1 . By Definition 2.4, C 1 is a subalgebra of R B pAq in H H YDpBq. We denote its H-action and coaction by a C 1 , δ C 1 . Using the isomorphismφ from Theorem 4.5, we define an H-action a C 2 and an H-coaction δ C 2 on C 2 :" Cent l A H pAq by a C 2 :"φa C 1 pId H bφ´1q, δ C 2 :" pId H bφqδ C 1φ´1 . " δ γφφ´1 " δ γ, and the compatibility with the H-actions is proved similarly.
The result below also follows from Theorem 4.5, cf. Equation 4.14.
Corollary 4.18. Assume that B " K-Mod for K a quasi-triangular Hopf algebra with braiding Ψ. Then, Z B pAq is the subalgebra Cent l A H pAq Ψ´1 of pA Hq Ψ´1 , which is a K-module algebra, given by The main results of this manuscript, Theorems 3.26 and 4.5, give generalizations of the results above. Namely, for an algebra A in a B-central monoidal category C, the B-center Z B pAq of A serves as a Morita invariant, and in the case when C " H-ModpBq, for some Hopf algebra H in B, we have that Z B pAq is the braided centralizer algebra Cent l A H pAq Ψ´1 in Z B pCq.

Braided Drinfeld doubles and Heisenberg doubles
Towards obtaining concrete examples of the results in the previous sections, we discuss here braided versions of useful algebraic constructions: the Drinfeld double and the Heisenberg double. Here, we restrict our attention to the case when C " H-ModpBq as in Example 3.2(3). Consider the following notation that we will use below and in the following sections.
Notation 5.1. Here, B " K-Mod for K a quasi-triangular Hopf algebra over k with R-matrix and its inverse denoted R p1q b R p2q and R p´1q b R p´2q , respectively. The braiding Ψ V,W for objects V, W in B is given by Ψ V,W pv b wq " pR p2q¨w q b pR p1q¨v q, @v P V, w P W.
Take H to be a Hopf algebra in B. We use sumless Sweedler notation ∆pbq :" b p1q b b p2q and ∆pdq :" d p1q b d p2q for b P H, d P K, and the coaction δ : We now recall material about braided Drinfeld doubles from [Lau19]; this construction is due to [Maj99] where it is called double bosonization. i.e. the left and right radical of x , y are trivial. Then, the braided Drinfeld double Drin K pH˚, Hq of H with respect to K and x , y is defined to be the Hopf algebra over k that is H˚b K b H as a k-vector space, and for b P H, c P H˚, d P K, has multiplication db " pd p1q¨b q d p2q , dc " pd p1q¨c q d p2q with counit the same as on H˚, K, H and extended multiplicatively, and with antipode Spdq " S K pdq, Spbq " S K pR p2q q pR p1q¨S H pbqq, Spcq " S K pR´p 1q q pR´p 2q¨S´1 H˚p cqq. for all c P H˚and v P V .
If H is finite-dimensional, then Φ is an equivalence of braided monoidal categories.
Example 5.4. If H " k, then Drin K pk˚, kq -K as k-Hopf algebras. Moreover, k-ModpBq is isomorphic to B as braided monoidal categories. So, Propositions 5.3 and 3.10 recover the fact that Z B pBq is isomorphic to B as braided monoidal categories as stated in Proposition 3.5(1). When K " k, the braided Drinfeld double is the usual Drinfeld double or quantum double as found, for example, in [Maj00, Theorem 7.1.1].
We saw in the previous section that the B-center of A P AlgpH-ModpBqq is isomorphic to the braided centralizer algebra of A in A H. So by continuing Example 3.18 and using the braided smash product algebras discussed in Section 4.1, consider the following special subclass of such algebras A H.
Definition 5.5. [Lau19, Example 3.10] For H a Hopf algebra in a braided monoidal category pB, Ψq, the braided smash product algebra˚H Ψ´1 H is called the braided Heisenberg double of H, and is denoted by Heis B pH,˚Hq.
Thus, Heis B pH,˚Hq is the K-module˚H b H with multiplication m Heis given by for a, b P˚H and g, h P H. Here x , y : H b˚H Ñ k is a nondegenerate Hopf algebra pairing, and R 1 , R 2 are copies of the universal R-matrix of K. If H is a k-Hopf algebra, then one recovers the Heisenberg double HeispHq from [Lu94, Definition 5.1].
We end with a basic example of a (braided) Heisenberg double for B " Vect k and compute an example of a Vect k -(or full-) center of an algebra in H-ModpVect k q below.
6. Example: Module algebras over u q psl 2 q In this section we provide an extended example of the material in the previous sections for the representation category of the finite-dimensional small quantum group u q psl 2 q. Recall Notation 5.1 and consider the following notation for the rest of this section.
Notation 6.1. Let n ě 3 be an integer , and let q a root of unity so that q 2 has order n.
‚ From [Maj00, Lemma 2.1.2 and Example 2.1.6], let K be the quasi-triangular Hopf algebra kZ n where Z n " xg | g n " 1y, with R-matrix and inverse given by ‚ Let B be the braided monoidal category K-Mod, which then, for g¨v " q 2|v| v with v belonging to an object in K-Mod, has braiding and inverse braiding ‚ Take H to be the Hopf algebra krxs{px n q in B, where along with εpx m q " δ m,0 and Spx m q " p´1q m q 2p m 2 q x m . Here, the K-action and the induced K-coaction on H are given by ‚ Next, take C to be the monoidal category H-ModpBq, which is equivalent to pH Kq-Mod. The smash product algebra H K is the Taft algebra T n pq´2q, i.e., the k-Hopf algebra T n pq´2q " kxg, xy{pg n´1 , x n , gx´q´2xgq, with ∆pgq " g b g, ∆pxq " g´1 b x`x b 1, εpgq " 1, εpxq " 0, Spgq " g´1, Spxq "´gx. That is, C is equivalent to T n pq´2q-Mod as a monoidal category.
‚ Pick A :" A γ " krus P AlgpCq with g¨u " q 2 u and x¨u " γ1 A , for γ P k.
Remark 6.2. We will show in Corollary 6.8 below that A γ"0 and A γ‰0 are not Morita equivalent as algebras in C, and to do so we compute their respective B-centers and employ Theorem 3.26. Even though this computation is tedious, it is, in a sense, more efficient to use the B-center Z B pAq " C l pR B pAqq [Theorem 3.24] as a Morita invariant rather than the full center ZpAq " C l pRpAqq [Dav10, Theorem 5.4]: Indeed, RpAq " H b K b A as a k-vector space [Dav12, Proposition 5.1], whereas R B pAq " H b A as a k-vector space [Theorem 3.13] and is a smaller algebra on which to do computations.
Recall that the functor R B : C Ñ Z B pCq exists in the setting above by Theorem 3.13 and it is lax monoidal. We compute the algebra R B pAq below. Lemma 6.3. Retain the notation above. Then the algebra R B pAq in H H YDpBq has the k-algebra presentation kxy, uy{py n , uy´q´2yuq.
As an object in B, the K-action on R B pAq is given by apg b yq " q´2y and apg b uq " q 2 u.
For the Yetter-Drinfeld structure, the H-action and H-coaction on R B pAq are given by Proof. As an object in H H YDpBq, we have that R B pAq " H b A. Here, we take H to be krys{py n q and denote the generators of H and of A in R B pAq by y :" y b 1 A and u :" 1 H b u, respectively. Moreover, take 1 :" 1 H b 1 A . The multiplication of R B pAq from Corollary 3.16 yields the relation uy " q´2yu in R B pAq; its other relations come from H.
The K-action a on R B pAq is the K-action on H and on A induced by the set-up of Notation 6.1.
The Yetter-Drinfeld structure pa R , δ R q on R B pAq is given in Theorem 3.13. We provide the details of one computation and leave the rest, including the verification of the Yetter-Drinfeld compatibility condition [Definition 3.6], to the reader: Proposition 6.4. Retain the notation above. Then, the B-center of A " A γ is For γ ‰ 0, we have that as an object in H H YDpBq, the braided commutative algebra Z B pA γ‰0 q has K-action, H-action, and H-coaction given by Now with Theorem 3.26 we arrive at the consequence below.
Corollary 6.8. The objects A γ"0 and A γ‰0 are Morita inequivalent as algebras in C.
Finally, we translate these results to u q psl 2 q-Mod. To do so, we return to braided Drinfeld doubles from Definition 5.2. Lemma 6.9. Consider the quasi-triangular k-Hopf algebra K " kZ n , with Z n " xg | g n " 1y, and take Hopf algebras H " krxs{px n q and H˚" krx˚s{ppx˚q n q in K-Mod as in Notation 6.1. Choose the nondegenerate pairing x , y : H˚b H Ñ k determined by xx˚, xy " 1 q´q´1 .
Then, the braided Drinfeld double Drin K pH˚, Hq of H with respect to K and the pairing above is generated by a group-like element g, a pg´1, 1q-skew primitive element x, and a pg´1, 1q-skew primitive element x˚, subject to relations: g n " 1, x n " px˚q n " 0, gx " q´2xg, gx˚" q 2 x˚g, x˚x´q 2 xx˚" 1 q´q´1 p1´g´2q.
Proof. We provide one computation and leave the rest to the reader. Note that Now, from Definition 5.2, consider the relation pR p´1q¨b p2q q pR p´2q¨c p1q q xc p2q , b p1q y " R p´1q c p2q b p1q q R p2q xR´p 2q¨c p1q , R p1q¨b p2q y, for b " x and c " x˚. The left-hand side is pR p´1q¨x q pR p´2q¨x˚q x1˚, 1y`pR p´1q¨1 q pR p´2q¨1˚q xx˚, xy " pg´1¨xqx˚`pg´1¨1q1˚xx˚, xy " q 2 xx˚`1 q´q´1 , and the right-side is R p´1q x˚x R p2q xR´p 2q¨1˚, R p1q¨1 y`R p´1q 1˚1 R p2q xR´p 2q¨x˚, R p1q¨x y " R p´1q x˚x R p2q xεpR´p 2q q, εpR p1q qy`g´1 1˚1 g´1 xx˚, xy " R p´1q εpR´p 2q q x˚x εpR p1q qR p2q x1, 1y`g´2 1 q´q´1 " x˚x`g´2 1 q´q´1 . The last equation holds as pId b εqR´1 " pε b idqR " 1. Thus, x˚x´q 2 xx˚" 1 q´q´1 p1´g´2q.
Now consider the small quantum group u q psl 2 q, for q a root of unity so that q 2 has order n for n ě 3 (e.g., as in [Kas95]). We take u q psl 2 q to be generated by indeterminates k, e, f , subject to relations k n " 1, e n " f n " 0, ke " q 2 ek, kf " q´2f k, ef´f e " k´k´1 q´q´1 , where ∆pkq " k b k, ∆peq " 1 b e`e b k, ∆pf q " k´1 b f`f b 1, εpkq " 1, εpeq " εpf q " 0.
(1) We have that u q psl 2 q is isomorphic to the braided Drinfeld double Drin K pH˚, Hq from Lemma 6.9.
(2) The Hopf subalgebra u q psl2 q (the negative Borel part) of u q psl 2 q generated by k and f is isomorphic to the Taft algebra T n pq´2q.
(3) We have that R B pA γ q from Lemma 6.3 is an algebra in u q psl 2 q-Mod via k¨y " q´2y, f¨y " p1´q 2 qy 2 , e¨y " 1 q´q´1 , k¨u " q 2 u, f¨u " p1´q´4qyu`γ, e¨u " 0.
(4) Moreover, Z B pA γ q from Proposition 6.4 is a commutative algebra in u q psl 2 q-Mod via k¨z " q 2 z, f¨z " γ, e¨z "´qγ´1z 2 . Proof.
(3) The action of k and of f follows from Lemma 6.3 using the isomorphism π from above. By Proposition 5.3 and Lemma 6.3, along with the pairing in Lemma 6.9, we have that x˚¨y " xx˚, y p´1q y y p0q " xx˚, 1 H y y`xx˚, yy 1 A " 1 q´q´1 , x˚¨u " xx˚, u p´1q y u p0q " xx˚, 1 H y u " 0. Now the conclusion holds by the isomorphism π from above.
(4) The action of k and of f follows from Proposition 6.4 and using π from above. By Propositions 5.3 and 6.4, along with the pairing in Lemma 6.9, we have that x˚¨z " xx˚, z p´1q y z p0q " γ´1p1´q 2 qq´2 p2q xx˚, yy z 2 " 1´q 2 γq 4 pq´q´1q z 2 "´1 γq 3 z 2 .
Namely, for δ R pzq in Proposition 6.4, we get that xx˚, y i y is nonzero only when i " 1. Now the conclusion holds by the isomorphism π above.

Example: Module algebras over the Sweedler Hopf algebra
We provide an example illustrating that the B-center of a parametrized family of algebras can be parameter-independent. Take charpkq ‰ 2, recall Notation 5.1, and consider the following notation.
Notation 7.1. For ξ P k, let pK, R ξ q be the Sweedler Hopf algebra K " T 2 p´1q " kxg, xy{pg 2´1 , x 2 , gx`xgq, with g grouplike and x being pg, 1q-skew-primitive, which is quasi-triangular with R-matrix for ξ P k. Take B " K-Mod, which has braiding Ψ B ξ pv, wq " pR p2q ξ¨w q b pR p1q ξ¨v q. Let C " H-ModpBq, with H " k a Hopf algebra in B. Take A γ to be krus P AlgpCq with g¨u "´u and x¨u " γ, for γ P k.
As discussed in Section 3.6, both C and its B-center Z B pCq are isomorphic to B, and moreover Ψ Z B pCq " Ψ B ": Ψ. Further, the B-center Z B pA γ q is actually the left center C l pA γ q. Now .
Using the fact that x¨u i " γu i´1 if i is odd and " 0 if i even, we see that mΨpu i buq "´u i`1`ξ γ 2 u i´1 if i is odd and " u i`1 if i even. Therefore, from the condition defining C l pA γ q, we have that λ 1 ξγ 2 " 0,´λ i`λi`2 ξγ 2 " λ i for i odd, and λ i is free for i even.
So, λ i " 0 for all i odd, and λ i is free for i even, no matter the choice of ξ and γ. Therefore, Proposition 7.2. The B-center of the parameterized family of T 2 p´1q-module algebras A γ is Z B pA γ q " C l pA γ q " kru 2 s with g¨u 2 " u 2 , x¨u 2 " 0, as a commutative algebra in C -B -Z B pCq.
8. On braided commutative module algebras over U q pgq and u q pgq In this section, we provide an avenue to generalize the results of Section 6 on braided commutative module algebras over u q psl 2 q to those over the quantized enveloping algebras U q pgq and u q pgq (as in settings ( †) and ( ‡) in the Introduction). To do so, we present the set-up of Notation 6.1 (for u q psl 2 q) in the context of U q pgq and u q pgq in Sections 8.1-8.4; here, we use the versions of quantum groups appearing in [CP95,BG02]. We end in Section 8.5 with proposing several possible directions for future research to continue this work.
8.1. The braided monoidal category B. First, fix a Cartan datum pI,¨q. That is, let I be a finite set, L " ZxIy be the lattice generated by I, and¨a symmetric bilinear form on L such that i¨i is even and a ij :" 2 i¨j i¨i P Z ď0 , for all i ‰ j. Let q be a free variable and F :" kpqq. For the free abelian group L " Zxg i | i P Iy, setting Rpg i , g j q " q i¨j , for i, j P I, defines a dual R-matrix on the group algebra p K :" FL. Now set B to be the braided monoidal category of p K-comodules with the braiding obtained from R.
8.2. The setting p:q for U q pgq in detail. Let g denote the semisimple Lie algebra associated to pI,¨q, and take H :" U q pn´q to be the negative nilpotent part of the quantum group U q pgq. We have that H is a Hopf algebra in B. As an algebra, U q pn´q is generated by primitive elements f i , for i P I, subject to the quantum Serre relations (as in (8.2) below for f i " F i ). Further, H is a p K-Yetter-Drinfeld module, with p K-action induced using R from the coaction δ, where For C " H-ModpBq, we have that C is equivalent to U q pg´q-Mod w and the relative monoidal center Z B pCq is equivalent to U q pgq-Mod lfw as a braided monoidal category; see [Lau20, Theorem 4.7]. As in [CP95, Section 9.1], the algebra U q pgq is generated by E i , F i , K˘1 i , for i P I, subject to relations for i ‰ j P I. Here, q i " q i¨i{2 , and`n m˘q " rnsq! rmsq!rn´msq! , for n ě m, where rns q " q n´q´n q´q´1 . The coproduct is determined by Example 8.3. By Example 4.19, we get that U q pn´q is a commutative algebra in U q pgq-Mod via 8.3. The setting p;q for u q pgq in detail. We abuse notation and assume here that q P k is a primitive n-th root of unity, where n ě 3 is an odd integer and is coprime to 3 if g contains a G 2 -factor. Note that q 2 is also a primitive n-th root of unity. We set q i :" q i¨i{2 . Now take H :" u q pn´q, to be the negative nilpotent part of the quantum group u q pgq, where u q pgq is the finite dimensional quotient of the specialization U q pgq by the relations E n i " 0, F n i " 0, K n i " 1, for i P I. Now H is a Hopf algebra in the braided monoidal category B (abusing notation) of K-modules, where K is the quotient of the free group p K by the relations g n i for i P I. For C " H-ModpBq, we have that C is equivalent to u q pg´q-Mod and the relative monoidal center Z B pCq is equivalent to u q pgq-Mod as a braided monoidal category; see [Lau20, Theorem 4.9]. Moreover, an action on u q pn´q defined by the same formulas as in Example 8.3 also makes u q pn´q a commutative algebra in u q pgq-Mod.
8.4. Module algebras over u q pg´q. Now we show how to generalize the module algebra A γ over the Taft algebra T n pq´2q from Notation 6.1 to the context of Section 8.3. Recall that T n pq´2q -u q psl2 q by Proposition 6.10(2). So, here, we consider module algebras over the negative Borel part u q pg´q of u q pgq; this is the subalgebra generated by K i , F i for i P I. Now for the braided monoidal category C in Section 8.3, let us consider the algebra A in C defined as follows. Take γ to be a collection of scalars pγ i q iPI in k, and let A :" A γ " kxu i | i P Iy be the free associative algebra with actions of K and of H " u q pn´q given by g i¨uj " q i¨j u j , f i¨uj " δ i,j γ i 1 A , for i, j P I.

8.5.
Questions. Continuing the work in the previous section, we ask Question 8.4. What is the B-center of the algebra A γ in Section 8.4? How is it presented as a braided commutative algebra in u q pgq-Mod?
In particular, one could consider the following problem.
Problem 8.5. By Example 3.18, the algebra A γ , with γ i " 1{pq i´q´1 i q, can be replaced with a quotient isomorphic to u q pn`q Ψ´1 . Compute the B-center of u q pn`q Ψ´1 as the centralizer of u q pn`q Ψ´1 in the braided Heisenberg double of u q pn´q from Definition 5.5.
Moreover, motivated by Corollary 6.8, we ask: Problem 8.6. What are conditions on the scalars γ " tγ i u iPI that distinguish Morita equivalence classes for the u q pg´q-module algebras A γ ?
One can also consider actions on other quotients of A γ , actions in the context of Section 8.2 above, or revisit related work mentioned at the beginning or end of the Introduction, for further directions of investigation.